SLAC Magnetic Measurements
Date: 06-30-1995
Time: 13:31:13

Magnet Name (Serial #): 000
Bar Code Number: 001
Project: PEP II High Energy Ring Sextupoles
Test Stand: IR8, S1
Measurement Coil: # 1
Operator: pdr
Run Number: 7
Comment: Check Repeatability

Standardization Currents (A):
   250.0    10.0   200.0    10.0   200.0    10.0   200.0
    10.0

Test Currents (A):
    25.0    50.0   100.0   150.0   200.0   150.0   100.0
    50.0    25.0



            INTEGRATED SEXTUPOLE STRENGTH VS CURRENT

Coil: Radius =  .044882  m, # Turns =  20 
Average: # Rotations/Measurement =  10 , # Measurements =  4 

  Imag    sigImag     SL     sigSL      SL/I   sigSL/I 
  (A)       (A)     (T/m)    (T/m)    (T/m/kA) (T/m/kA)
--------+--------  --------+--------  --------+--------
  25.586    0.003   5.22818  0.00162  204.3379   0.0670
  50.572    0.005  10.19949  0.00266  201.6815   0.0559
 100.565    0.003  20.23740  0.00618  201.2373   0.0617
 150.547    0.005  30.24512  0.00632  200.9012   0.0424
 200.548    0.007  40.15477  0.00461  200.2250   0.0239
 150.554    0.004  30.51589  0.01286  202.6906   0.0856
 100.570    0.006  20.48722  0.00356  203.7111   0.0373
  50.566    0.002  10.40361  0.00260  205.7416   0.0519
  25.580    0.002   5.36461  0.00083  209.7149   0.0365


           SUMMARY OF THE CALCULATIONS AND CONVENTIONS USED


Field Expansion:
The expansion of the radial and azimuthal field in polar
coordinates is
 Br(r,th) = Sum Bn (r/rref)^(n-1) cos(n(th-thspole))
 Bth(r,th) = -Sum Bn (r/rref)^(n-1) sin(n(th-thspole))
Our convention is to set rref = Rcoil.
Thspole is the angle of the first magnetic south pole
with respect to the horizontal, measured ccw by a shaft
encoder in the system.

Coil Voltage:
The coil voltage from each field harmonic is
 Vn(th) = Nturns * velocity * Brn(Rcoil,th) * L
L is the magnet effective length, or BLn is the integrated
field strength of the n'th harmonic.
At the coil radius, the radial field as a function of angle is,
 Brn(Rcoil,th) * L = BLn * cos(n*(th - thspole))
The coil voltage is
 Vn(th) = Nturns * velocity * BLn * cos(n*(th - thspole))
        = Nturns * Rcoil * ang_freq * BLn * cos(n*(th - THspole))
An FFT of the coil voltage gives Vn and PhiVn according to the formula
 Vn(i) = Vn * cos(n*2pi*i/N + PhiVn)

Multipole Field Calculations:
To find the multipole field magnitudes and phases,
the measured voltage harmonics are related to their values
calculated from the field harmonics:
 Nturns * Rcoil * ang_freq * BLn = Vn
 -n * thspole = PhiVn
Or,
 BLn = Vn / (Nturns * Rcoil * ang_freq)
 thspole = -PhiVn / n

Harmonic Strength Ratios:
The main field, denoted by capital N, is the field harmonic
with the largest strength at the coil radius.
The field strength ratio is defined by
 Rn = BLn / BLN
It gives the ratio of each harmonic field strength to the
main field strength at the coil radius.

Calculation Of SL:
The sextupole strength S is defined by the vertical
field on the x-axis (median plane).
It is the quadratic term in the Taylor expansion.
 By(x) = 1/2 S x^2
On the x-axis, By(x) = Bth(r=x,th=0)
From the expression for Bth above,
 By(x) = - B3 (x/rref)^2 sin(-3 thspole)
Take rref = Rcoil, thspole = pi/6, then at x=Rcoil,
 By(Rcoil) = B3 = 1/2 S Rcoil^2
So,
 S = 2 B3 / Rcoil^2
The integrated sextupole strength is
 SL = 2 BL3 / Rcoil^2

Calculation Of The Sextupole Center:
In the sextupole's frame,
 Bx' = S * x' * y'
 By' = 1/2 * S * (x'^2 - y'^2).
In the coil's frame (unprimed frame) the magnetic center is at (x0, y0).
In the coil's frame,
 Bx = S * (x - x0) * (y - y0)
 By = 1/2 * S * [(x - x0)^2 - (y - y0)^2].
The magnetic center can be found in terms of the measured
quadrupole field.  Compare Br and Btheta on the x-axis to
 Bxquad = -S * (x0 * y + y0 * x)
 Byquad = -S * (x0 * x - y0 * y).
Evaluate at x = Rcoil, y = 0 and compare to
 Br(theta = 0) = Bx, Btheta(theta = 0) = By.
In terms of the measured integrated strengths, this gives
 Xcenter = x0 = - (1/(SL * Rcoil)) * BL2 * sin(2 * THspole2)
 Ycenter = y0 = - (1/(SL * Rcoil)) * BL2 * cos(2 * THspole2)